Graduate Lectures in Mathematics (9/13)

Wednesday, November 15, 2023

Sebastian Baader

Prof. Sebastian Baader

Dr. Annina Iseli, EPFL, und Prof. Felix Schlenk, Université de Neuchâtel

The purpose of these traditional lectures is to present recent important developments in mathematics to a large audience of graduate students and young researchers. The lecturers are chosen as experts in their field, with demonstrated pedagogical talent.

Preliminary schedule; to be confirmed with the second announcement

Wednesday, 13h15-14-45: Dr. Annina Iseli

Title: The dynamics of rational maps on the Riemann sphere

Abstract: Complex dynamics has been a very active research area in recent years. Its main goal is understanding the orbits of points under iterated applications of holomorphic self-maps of a complex surface or manifold - most classically the Riemann sphere for which the class of holomorphic maps coincides with the class of rational maps. The foundational results in the field are largely based on complex analytic methods. They include for example the description of the geometry of the set of points with stable resp. unstable orbits (Fatou and Julia set) and of the parameter spaces of specific classes of rational maps (Mandelbrot sets). A more recent approach, pioneered by Thurston, takes a more topological perspective and targets in particular rigidity results for rational maps. In the first part of this course, we will cover the base theory for both viewpoints, analytic and topological. In the second part, we will study selected recent results in the field.

The prerequisites for this course are a solid base knowledge in complex analysis, low dimensional topology, and differential geometry.

Wednesday, 15h00-16h30: Prof. Dr. Felix Schlenk

Title: Symplectic embedding problems Symplectic embedding problems

Abstract: Symplectic geometry is a geometry that arose from the geometrization of celestial mechanics, but nowadays is at the cross-road of many branches of mathematics: of virtually all geometries, low-dimensional topology, mathematical physics, PDEs, etc.

This geometry is hard to "feel", since in contrast to Riemannian geometry there are no local invariants. One way to learn something about this geometry is by making experiments that lead to numbers: Given a simple subset U of R^2n such as finitely many balls, or an ellipsoid, what is the largest scaling \lambda U that symplectically embeds into a given symplectic manifold?

Answering such questions leads to unexpected connections with other fields, such as Diophantine equations, Markov numbers, isoperimetric problems in lattices, degenerations or almost toric fibration of the complex projective plane, etc.

The prerequisites for this course are minimal (a bachelor in Maths would do), but some knowledge in differential geometry is useful.

University of Bern, building ExWi, room B78

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